Building a Möbius Bracelet Using Safety Pins: A Problem of Modular Arithmetic and Staggered Positions

Abstract:

This article reports on the resolution of a mathematical problem that emerged when two ideas were brought
together. The first idea consists of a method for constructing a decorated bracelet made with safety pins that are
strung together at both ends, creating a band. The other is suggested by the word band: why not introduce a twist
and make the bracelet a Möbius band? As Isaksen and Petrofsky demonstrated in their paper [1] discussing the
knitting of a Möbius band, the endless nature of the band’s single face and edge introduces an additional design
constraint, particularly if the connection is to appear seamless. To make the creation appear seamless, the
decoration applied to the design must itself be regular, as this helps the eye travel along the endless length. The
paper discusses the mathematical and practical constraints of this result for a design that uses a repeating pattern
throughout the band, first in the standard design, then in the Möbius bracelet. This resolution involves some simple
modular arithmetic and an unusual way to lay out the pins in preparation for their being strung together.